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Derivative and Integral Cheat Sheet, Cheat Sheet of Calculus

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Common Derivatives and Integrals
Derivatives
Basic Properties/Formulas/Rules
( )
( )
( )
dcf x cf x
dx
=
, c is any constant.
( ) ( )
( )
( ) ( )
fxgx fxgx
′′
±=±
( )
1nn
dx nx
dx
=
, n is any number.
( )
0
dc
dx =
, c is any constant.
( )
fg f g fg
′′
= +
(Product Rule)
2
f f g fg
gg
′′

=


(Quotient Rule)
( )
( )
( )
( )
( )
( )
df gx f gx g x
dx ′′
=
(Chain Rule)
( )
( )
( )
( )
gx gx
dgx
dx
=ee
Common Derivatives
Polynomials
( )
0
dc
dx =
( )
1
dx
dx =
( )
dcx c
dx =
( )
1nn
dx nx
dx
=
( )
1nn
dcx ncx
dx
=
Trig Functions
( )
sin cos
dxx
dx =
( )
cos sin
dxx
dx =
( )
2
tan sec
dxx
dx =
( )
sec sec tan
dx xx
dx =
( )
csc csc cot
dx xx
dx =
( )
2
cot csc
dxx
dx =
Inverse Trig Functions
( )
1
2
1
sin 1
dx
dx x
=
( )
1
2
1
cos 1
dx
dx x
=
( )
1
2
1
tan 1
dx
dx x
=+
( )
1
2
1
sec 1
dx
dx xx
=
( )
1
2
1
csc 1
dx
dx xx
=
( )
1
2
1
cot 1
dx
dx x
= +
Exponential/Logarithm Functions
( )
( )
ln
xx
daaa
dx =
( )
xx
d
dx =ee
( )
( )
1
ln , 0
dxx
dx x
= >
( )
1
ln , 0
dxx
dx x
=
( )
( )
1
log , 0
ln
a
dxx
dx x a
= >
Hyperbolic Trig Functions
( )
sinh cosh
dxx
dx =
( )
cosh sinh
dxx
dx =
( )
2
tanh sech
dxx
dx =
( )
sech sech tanh
dx xx
dx =
( )
csch csch coth
dx xx
dx =
( )
2
coth csch
dxx
dx =
pf3
pf4
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Derivatives

Basic Properties/Formulas/Rules

d cf x cf x dx

= ′ , c is any constant. ( ) ( )

( f^ x^ g^ x^ ) f^ (^ x^ )^ g^ (^ x)

d (^) n n 1 x nx dx

= , n is any number. ( ) 0

d c dx

= , c is any constant.

( f g^ ) f^ g^ f g

′ (^) = ′ + ′ – (Product Rule) 2

f f g f g

g g

  • (Quotient Rule)

( (^ (^ ))) (^ (^ ))^ (^ )

d f g x f g x g x dx

= ′^ ′ (Chain Rule)

( )

( ) (^ )^

d (^) g x g x( ) g x dx

e = ′ e ( ( ))

ln

d g^ x g x dx g x

Common Derivatives

Polynomials

d c dx

d x dx

d cx c dx

d (^) n n 1 x nx dx

d (^) n n 1 cx ncx dx

Trig Functions

( sin^ ) cos

d x x dx

= ( cos ) sin

d x x dx

2 tan sec

d x x dx

( sec^ ) sec^ tan

d x x x dx

= ( csc ) csc cot

d x x x dx

2 cot csc

d x x dx

Inverse Trig Functions

1 2

sin 1

d x dx (^) x

1 2

cos 1

d x dx (^) x

− = − −

1 2

tan 1

d x dx x

1 2

sec 1

d x dx (^) x x

1 2

csc 1

d x dx (^) x x

− = − −

1 2

cot 1

d x dx x

− = −

Exponential/Logarithm Functions

( ) ln(^ )

d (^) x x a a a dx

d x x

dx

e = e

ln , 0

d x x dx x

ln , 0

d x x dx x

log , 0 ln

a

d x x dx x a

Hyperbolic Trig Functions

( sinh^ ) cosh

d x x dx

= ( cosh ) sinh

d x x dx

2 tanh sech

d x x dx

( sech^ ) sech^ tanh

d x x x dx

= − ( csch ) csch coth

d x x x dx

2 coth csch

d x x dx

Integrals

Basic Properties/Formulas/Rules

∫ cf^ (^ x dx)^ =c^ ∫ f^ (^ x dx) ,^ c^ is a constant.^ ∫ f^ (^ x^ )^ ±^ g^ (^ x dx)^ =^ ∫ f^ (^ x dx)^ ±∫g^ (^ x dx)

( ) ( ) ( ) ( )

b (^) b

a a ∫ f^ x dx^ =^ F^ x^ =^ F b^ −F^ a where^ F^ (^ x^ )^ =^ ∫f^ (^ x dx)

( ) ( )

b b

a a ∫ cf^ x dx^ =c^ ∫ f^ x dx,^ c^ is a constant.^ (^ )^ (^ )^ (^ )^ (^ )

b b b

a a a ∫ f^ x^ ±^ g^ x dx^ =^ ∫ f^ x dx^ ±∫ g^ x dx

( ) 0

a

a ∫ f^ x dx^ = (^ )^ (^ )

b a

a b ∫ f^ x dx^ = −∫ f^ x dx

( ) ( ) ( )

b c b

a a c ∫ f^ x dx^ =^ ∫ f^ x dx^ +∫ f^ x dx (^ )

b

a ∫ c dx^ =^ c b^ −a

If f (^) ( x (^) ) ≥ 0 on a ≤ x ≤ bthen (^) ( ) 0

b

a ∫ f^ x dx^ ≥

If f (^) ( x (^) ) ≥ g (^) ( x)on a ≤ x ≤ bthen (^) ( ) ( )

b b

a a ∫ f^ x dx^ ≥∫ g^ x dx

Common Integrals

Polynomials

dx = x +c ∫

k dx = k x +c ∫

n n x dx x c n n

= + ≠ −

dx lnx c x

1 x dx lnx c

− = + ∫

n n x dx x c n n

− − + = + ≠ − +

dx lnax b c ax b a

p p p q q q q p q

q x dx x c x c p q

+^ + = + = +

Trig Functions

∫ cos^ u du^ =^ sinu^ +c ∫ sin^ u du^ = −^ cosu^ +c

2 ∫sec^ u du^ =^ tanu^ +c

∫ sec^ u^ tan^ u du^ =^ secu^ +c ∫ csc^ u^ cot^ udu^ = −^ cscu^ +c

2 ∫csc^ u du^ = −^ cotu^ +c

∫ tan^ u du^ =^ ln secu^ +c ∫cot^ u du^ =^ ln sinu^ +c

∫ sec^ u du^ =^ ln sec^ u^ +^ tanu^ +c (^ )

sec sec tan ln sec tan 2

∫^ u du^ =^ u^ u^ +^ u^ +^ u^ +c

∫ csc^ u du^ =^ ln csc^ u^ −^ cotu^ +c (^ )

csc csc cot ln csc cot 2

∫^ u du^ =^ −^ u^ u^ +^ u^ −^ u^ +c

Exponential/Logarithm Functions

u u ∫ e^ du^ =^ e^ +c ln

u u a a du c a

∫ =^ + ∫ln^ u du^ =^ u^ ln(^ u^ )−^ u^ +c

sin ( ) (^2 2) ( sin ( ) cos( ))

au au bu du a bu b bu c a b

e e ( 1 )

u u ∫ u^ e^ du^ =^ u^ −^ e +c

cos ( ) (^2 2) ( cos ( ) sin( ))

au au bu du a bu b bu c a b

e e

ln ln ln

du u c u u

Trig Substitutions

If the integral contains the following root use the given substitution and formula.

2 2 2 2 2 sin and cos 1 sin

a a b x x b

2 2 2 2 2 sec and tan sec 1

a b x a x b

2 2 2 2 2 tan and sec 1 tan

a a b x x b

Partial Fractions

If integrating

( )

( )

P x dx Q x

where the degree (largest exponent) of P x( )is smaller than the

degree of Q x( ) then factor the denominator as completely as possible and find the partial

fraction decomposition of the rational expression. Integrate the partial fraction

decomposition (P.F.D.). For each factor in the denominator we get term(s) in the

decomposition according to the following table.

Factor in Q x( ) Term in P.F.D Factor in Q x( ) Term in P.F.D

ax + b

A

ax + b

( )

k ax + b ( ) ( )

1 2 2

k k

A A A

ax b (^) ax b ax b

2 ax + bx + c 2

Ax B

ax bx c

( )

2 k ax + bx + c ( )

1 1 (^2 )

k k k

A x B A x B

ax bx c (^) ax bx c

Products and (some) Quotients of Trig Functions

sin cos

n m ∫ x^ x dx

1. If n is odd. Strip one sine out and convert the remaining sines to cosines using 2 2 sin x = 1 − cos x, then use the substitution u =cosx 2. If m is odd. Strip one cosine out and convert the remaining cosines to sines

using

2 2 cos x = 1 − sin x, then use the substitution u =sinx

3. If n and m are both odd. Use either 1. or 2. 4. If n and m are both even. Use double angle formula for sine and/or half angle

formulas to reduce the integral into a form that can be integrated.

tan sec

n m ∫ x^ x dx

1. If n is odd. Strip one tangent and one secant out and convert the remaining

tangents to secants using

2 2 tan x = sec x− 1 , then use the substitution u =secx

2. If m is even. Strip two secants out and convert the remaining secants to tangents

using

2 2 sec x = 1 + tan x, then use the substitution u =tanx

3. If n is odd and m is even. Use either 1. or 2. 4. If n is even and m is odd. Each integral will be dealt with differently.

Convert Example : (^) ( ) ( )

6 2 3 2 3 cos x = cos x = 1 −sin x